inverse function theorem (topological spaces)

Let $X$ and $Y$ be topological spaces, with $X$ compact and $Y$ Hausdorff. Suppose $f:X\to Y$ is a continuous bijection. Then $f$ is a homeomorphism, i.e. $f^{-1}$ is continuous.

Note if $Y$ is a metric space, then it is Hausdorff, and the theorem holds.

Title	inverse function theorem (topological spaces)
Canonical name	InverseFunctionTheoremtopologicalSpaces
Date of creation	2013-03-22 13:25:04
Last modified on	2013-03-22 13:25:04
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	9
Author	mathcam (2727)
Entry type	Theorem
Classification	msc 54C05
Related topic	Compact